Question: Simplify; express your answer in exponential form. Assume $k\neq 0, z\neq 0$. $\dfrac{{(k^{3}z^{4})^{4}}}{{(k^{4}z)^{-3}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{3}z^{4})^{4} = (k^{3})^{4}(z^{4})^{4}}$ On the left, we have ${k^{3}}$ to the exponent ${4}$ . Now ${3 \times 4 = 12}$ , so ${(k^{3})^{4} = k^{12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{3}z^{4})^{4}}}{{(k^{4}z)^{-3}}} = \dfrac{{k^{12}z^{16}}}{{k^{-12}z^{-3}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{12}z^{16}}}{{k^{-12}z^{-3}}} = \dfrac{{k^{12}}}{{k^{-12}}} \cdot \dfrac{{z^{16}}}{{z^{-3}}} = k^{{12} - {(-12)}} \cdot z^{{16} - {(-3)}} = k^{24}z^{19}$